Schreiber
integration of ∞-Lie algebroid valued differential forms

An integration of ∞-Lie algebroid valued differential forms ω:Π inf(X)𝔞\omega : \Pi^{inf}(X) \to \mathfrak{a} is an extension ω\int \omega from the infinitesimal path ∞-groupoid? Π inf(X)\Pi^{inf}(X) to the finite path ∞-groupoid Π(X)\Pi(X)

Π inf(X) ω 𝔞 Π(x) ω A \array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(x) &\stackrel{\int \omega}{\to}& A }

after injecting the ∞-Lie algebroid 𝔞\mathfrak{a} into one of its corresponding ∞-Lie groupoids AA.

Contents

Idea

Assume a context given by a smooth (∞,1)-topos with its notion of ∞-Lie theory.

For AA some ∞-Lie groupoid and UCU \in C a test domain, a morphism ϕ:Π inf(U)A\phi : \Pi^{inf}(U) \to A from the infinitesimal path ∞-groupoid? factors – by definition of ∞-Lie algebroid – through the ∞-Lie algebroid 𝔞\mathfrak{a} of AA

Π inf(U)ω𝔞A \Pi^{inf}(U) \stackrel{\omega}{\to} \mathfrak{a} \hookrightarrow A

The first morphism here constitutes the set of ∞-Lie algebroid valued differential forms on UU that is encoded by ϕ\phi.

We want to specify what it means to integrate these forms.

Whereas ω\omega colors infinitesimal intervals on UU by infinitesimal morphisms in AA, its integration should be a rule that labels finite intervals by finite morphisms, such that the restriction to any infinitesimal interval within a finite interval reproduces the assignment of omgea\omgea.

Formally, this desideratum clearly means that integration ω\int \omega of ω\omega is an extension

Π inf(U) ω A ω Π(U) \array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow & \nearrow_{\int \omega} \\ \Pi(U) }

of ω\omega through the inclusion Π inf(U)Π(U)\Pi^{inf}(U) \hookrightarrow \Pi(U) of the infinitesimal path ∞-groupoid? in the path ∞-groupoid.

More generally, we can ask for a relative integration : we may have a situation where a coloring of finite intervals

Π(U)B \Pi(U) \to B

is already given, but infinitesimally there is also a lift specified of this through some map ABA \to B, i.e. a diagram

Π inf(U) ω A Π(U) B \array{ \Pi^{inf}(U) &\stackrel{\omega}{\to}& A \\ \downarrow && \downarrow \\ \Pi(U) &\to& B }

Integration should exist when the map ABA \to B in principle admits such a lift and should be given by that lift. It should be unique up to equivalence.

Details

We discuss a case in which integration of \infty-Lie algebroid valued forms always exsist.

Let 𝔞\mathfrak{a} be an ∞-Lie algebroid.

Write for short 𝔞 #:=𝔞 # inf\mathfrak{a}_# := \mathfrak{a}_{#^{inf}} in the following for the image of 𝔞\mathfrak{a} under the right adjoint of the infinitesimal path ∞-groupoid? functor Π inf\Pi^{inf}. And write 𝔞 #0\mathfrak{a}_{# 0} for the corresponding presheaf of 0-cells.

By the reasoning an ∞-Lie differentiation and integration we have

  • the \infty-Lie algebroid of 𝔞\mathfrak{a} is 𝔞\mathfrak{a} itself:

    Lie(𝔞):=Π inf(𝔞 #0)=𝔞. Lie(\mathfrak{a}) := \Pi^{inf}(\mathfrak{a}_{# 0}) = \mathfrak{a} \,.
  • the ∞-Lie groupoid integrating 𝔞\mathfrak{a} is

    A=Π(𝔞 #0). A = \Pi(\mathfrak{a}_{# 0}) \,.

For XX simplicially discrete, every morphism

ω:Π inf(X)𝔞 \omega : \Pi^{inf}(X) \to \mathfrak{a}

corresponding by adjunction to a morphism

ω˜:X𝔞 # 0 \tilde \omega : X \to \mathfrak{a}_{#_0}

induces a morphism

ω:=Π(ω˜):Π(X)Π(𝔞 #0)=A \int \omega := \Pi(\tilde \omega) : \Pi(X) \to \Pi(\mathfrak{a}_{# 0}) = A

that fits into the diagram

Π inf(X) ω Π inf(𝔞 #0) = 𝔞 Π(X) ω Π(𝔞 #0) = A. \array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \Pi^{inf}(\mathfrak{a}_{# 0}) &=& \mathfrak{a} \\ \downarrow && \downarrow && \downarrow \\ \Pi(X) &\stackrel{\int \omega}{\to}& \Pi(\mathfrak{a}_{# 0}) & = & A } \,.

Revised on May 29, 2012 22:04:00 by Andrew Stacey (129.241.15.200)